Discount advertisement in social platform: algorithm and robust analysis

As a marketing strategy, discount promotion is adopted by plenty of companies, which can be regarded as a variant of the previous Profit Maximization (PM) problem. Based on a discount-based marketing scenario, we propose a Profit Maximization with Discount Advertisement (PMDA) problem. Then, we show that the objective function of PMDA is submodular but not monotone, which can be categorized as an instance of Unconstrained Submodular Maximization problem. Even that similar problem has been studied before, the approximation performance is not satisfactory. Learned from the latest results, we combine the idea of greedy algorithm and randomized double greedy algorithm to solve our problem, which overcomes the shortcomings of both and obtains a more acceptable approximation ratio. It can be used as a general algorithmic framework. Moreover, the existing researches about PM only considered to maximize total profit based on certain diffusion probabilities. Because of the uncertainty of diffusion probabilities, we study the robustness of PMDA and propose Robust-PMDA problem further. It aims to acquire the maximum worst ratio between the profit of selected seed set and the optimal seed set. To solve the Robust-PMDA, we design LU-PMDASolver algorithm first, and then, we propose P-UniSampling algorithm to improve the robustness by reducing the uncertainly of diffusion probabilities, which is implemented by the technique of uniform sampling. Finally, the correctness and performance of our proposed algorithms are verified by conducting experiments on real-world social networks.